LMFDB - Embedded newform 55.3.d.b.54.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [55,3,Mod(54,55)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(55, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("55.54");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 55 = 5 \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	55.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$1.49864145398$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			54.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	55.54

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607 q^{2} +1.00000 q^{4} +5.00000 q^{5} +4.47214 q^{7} +6.70820 q^{8} +9.00000 q^{9} -11.1803 q^{10} -11.0000 q^{11} +22.3607 q^{13} -10.0000 q^{14} -19.0000 q^{16} -31.3050 q^{17} -20.1246 q^{18} +5.00000 q^{20} +24.5967 q^{22} +25.0000 q^{25} -50.0000 q^{26} +4.47214 q^{28} +18.0000 q^{31} +15.6525 q^{32} +70.0000 q^{34} +22.3607 q^{35} +9.00000 q^{36} +33.5410 q^{40} -84.9706 q^{43} -11.0000 q^{44} +45.0000 q^{45} -29.0000 q^{49} -55.9017 q^{50} +22.3607 q^{52} -55.0000 q^{55} +30.0000 q^{56} -102.000 q^{59} -40.2492 q^{62} +40.2492 q^{63} +41.0000 q^{64} +111.803 q^{65} -31.3050 q^{68} -50.0000 q^{70} -78.0000 q^{71} +60.3738 q^{72} +4.47214 q^{73} -49.1935 q^{77} -95.0000 q^{80} +81.0000 q^{81} +165.469 q^{83} -156.525 q^{85} +190.000 q^{86} -73.7902 q^{88} +2.00000 q^{89} -100.623 q^{90} +100.000 q^{91} +64.8460 q^{98} -99.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 10 q^{5} + 18 q^{9} - 22 q^{11} - 20 q^{14} - 38 q^{16} + 10 q^{20} + 50 q^{25} - 100 q^{26} + 36 q^{31} + 140 q^{34} + 18 q^{36} - 22 q^{44} + 90 q^{45} - 58 q^{49} - 110 q^{55} + 60 q^{56}+ \cdots - 198 q^{99}+O(q^{100}) $ 2 * q + 2 * q^4 + 10 * q^5 + 18 * q^9 - 22 * q^11 - 20 * q^14 - 38 * q^16 + 10 * q^20 + 50 * q^25 - 100 * q^26 + 36 * q^31 + 140 * q^34 + 18 * q^36 - 22 * q^44 + 90 * q^45 - 58 * q^49 - 110 * q^55 + 60 * q^56 - 204 * q^59 + 82 * q^64 - 100 * q^70 - 156 * q^71 - 190 * q^80 + 162 * q^81 + 380 * q^86 + 4 * q^89 + 200 * q^91 - 198 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/55\mathbb{Z}\right)^\times$.

$n$	$12$	$46$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.23607	−1.11803	−0.559017	−	0.829156i	$-0.688821\pi$
$2$	−2.23607	−1.11803	−0.559017	+	0.829156i	$0.688821\pi$
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	1.00000	0.250000
$5$	5.00000	1.00000
$5$	5.00000	1.00000
$6$	0	0
$7$	4.47214	0.638877	0.319438	−	0.947607i	$-0.396506\pi$
$7$	4.47214	0.638877	0.319438	+	0.947607i	$0.396506\pi$
$8$	6.70820	0.838525
$9$	9.00000	1.00000
$10$	−11.1803	−1.11803
$11$	−11.0000	−1.00000
$11$	−11.0000	−1.00000
$12$	0	0
$13$	22.3607	1.72005	0.860026	−	0.510250i	$-0.170447\pi$
$13$	22.3607	1.72005	0.860026	+	0.510250i	$0.170447\pi$
$14$	−10.0000	−0.714286
$15$	0	0
$16$	−19.0000	−1.18750
$17$	−31.3050	−1.84147	−0.920734	−	0.390191i	$-0.872409\pi$
$17$	−31.3050	−1.84147	−0.920734	+	0.390191i	$0.872409\pi$
$18$	−20.1246	−1.11803
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	5.00000	0.250000
$21$	0	0
$22$	24.5967	1.11803
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	25.0000	1.00000
$26$	−50.0000	−1.92308
$27$	0	0
$28$	4.47214	0.159719
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	18.0000	0.580645	0.290323	−	0.956929i	$-0.406237\pi$
$31$	18.0000	0.580645	0.290323	+	0.956929i	$0.406237\pi$
$32$	15.6525	0.489140
$33$	0	0
$34$	70.0000	2.05882
$35$	22.3607	0.638877
$36$	9.00000	0.250000
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	33.5410	0.838525
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	−84.9706	−1.97606	−0.988030	−	0.154262i	$-0.950700\pi$
$43$	−84.9706	−1.97606	−0.988030	+	0.154262i	$0.950700\pi$
$44$	−11.0000	−0.250000
$45$	45.0000	1.00000
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	−29.0000	−0.591837
$50$	−55.9017	−1.11803
$51$	0	0
$52$	22.3607	0.430013
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	−55.0000	−1.00000
$56$	30.0000	0.535714
$57$	0	0
$58$	0	0
$59$	−102.000	−1.72881	−0.864407	−	0.502793i	$-0.832306\pi$
$59$	−102.000	−1.72881	−0.864407	+	0.502793i	$0.832306\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	−40.2492	−0.649181
$63$	40.2492	0.638877
$64$	41.0000	0.640625
$65$	111.803	1.72005
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	−31.3050	−0.460367
$69$	0	0
$70$	−50.0000	−0.714286
$71$	−78.0000	−1.09859	−0.549296	−	0.835628i	$-0.685104\pi$
$71$	−78.0000	−1.09859	−0.549296	+	0.835628i	$0.685104\pi$
$72$	60.3738	0.838525
$73$	4.47214	0.0612621	0.0306311	−	0.999531i	$-0.490248\pi$
$73$	4.47214	0.0612621	0.0306311	+	0.999531i	$0.490248\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−49.1935	−0.638877
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−95.0000	−1.18750
$81$	81.0000	1.00000
$82$	0	0
$83$	165.469	1.99360	0.996801	−	0.0799187i	$-0.0254661\pi$
$83$	165.469	1.99360	0.996801	+	0.0799187i	$0.0254661\pi$
$84$	0	0
$85$	−156.525	−1.84147
$86$	190.000	2.20930
$87$	0	0
$88$	−73.7902	−0.838525
$89$	2.00000	0.0224719	0.0112360	−	0.999937i	$-0.496423\pi$
$89$	2.00000	0.0224719	0.0112360	+	0.999937i	$0.496423\pi$
$90$	−100.623	−1.11803
$91$	100.000	1.09890
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	64.8460	0.661694
$99$	−99.0000	−1.00000
$100$	25.0000	0.250000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	150.000	1.44231
$105$	0	0
$106$	0	0
$107$	−156.525	−1.46285	−0.731424	−	0.681923i	$-0.761144\pi$
$107$	−156.525	−1.46285	−0.731424	+	0.681923i	$0.761144\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	122.984	1.11803
$111$	0	0
$112$	−84.9706	−0.758666
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	201.246	1.72005
$118$	228.079	1.93287
$119$	−140.000	−1.17647
$120$	0	0
$121$	121.000	1.00000
$122$	0	0
$123$	0	0
$124$	18.0000	0.145161
$125$	125.000	1.00000
$126$	−90.0000	−0.714286
$127$	−31.3050	−0.246496	−0.123248	−	0.992376i	$-0.539331\pi$
$127$	−31.3050	−0.246496	−0.123248	+	0.992376i	$0.539331\pi$
$128$	−154.289	−1.20538
$129$	0	0
$130$	−250.000	−1.92308
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	−210.000	−1.54412
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	22.3607	0.159719
$141$	0	0
$142$	174.413	1.22826
$143$	−245.967	−1.72005
$144$	−171.000	−1.18750
$145$	0	0
$146$	−10.0000	−0.0684932
$147$	0	0
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	−281.745	−1.84147
$154$	110.000	0.714286
$155$	90.0000	0.580645
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	78.2624	0.489140
$161$	0	0
$162$	−181.122	−1.11803
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	−370.000	−2.22892
$167$	219.135	1.31218	0.656092	−	0.754681i	$-0.272208\pi$
$167$	219.135	1.31218	0.656092	+	0.754681i	$0.272208\pi$
$168$	0	0
$169$	331.000	1.95858
$170$	350.000	2.05882
$171$	0	0
$172$	−84.9706	−0.494015
$173$	237.023	1.37008	0.685038	−	0.728507i	$-0.259786\pi$
$173$	237.023	1.37008	0.685038	+	0.728507i	$0.259786\pi$
$174$	0	0
$175$	111.803	0.638877
$176$	209.000	1.18750
$177$	0	0
$178$	−4.47214	−0.0251244
$179$	−38.0000	−0.212291	−0.106145	−	0.994351i	$-0.533851\pi$
$179$	−38.0000	−0.212291	−0.106145	+	0.994351i	$0.533851\pi$
$180$	45.0000	0.250000
$181$	−342.000	−1.88950	−0.944751	−	0.327788i	$-0.893697\pi$
$181$	−342.000	−1.88950	−0.944751	+	0.327788i	$0.893697\pi$
$182$	−223.607	−1.22861
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	344.354	1.84147
$188$	0	0
$189$	0	0
$190$	0	0
$191$	338.000	1.76963	0.884817	−	0.465939i	$-0.154283\pi$
$191$	338.000	1.76963	0.884817	+	0.465939i	$0.154283\pi$
$192$	0	0
$193$	−31.3050	−0.162202	−0.0811009	−	0.996706i	$-0.525844\pi$
$193$	−31.3050	−0.162202	−0.0811009	+	0.996706i	$0.525844\pi$
$194$	0	0
$195$	0	0
$196$	−29.0000	−0.147959
$197$	−84.9706	−0.431323	−0.215661	−	0.976468i	$-0.569191\pi$
$197$	−84.9706	−0.431323	−0.215661	+	0.976468i	$0.569191\pi$
$198$	221.371	1.11803
$199$	178.000	0.894472	0.447236	−	0.894416i	$-0.352408\pi$
$199$	178.000	0.894472	0.447236	+	0.894416i	$0.352408\pi$
$200$	167.705	0.838525
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	−424.853	−2.04256
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	350.000	1.63551
$215$	−424.853	−1.97606
$216$	0	0
$217$	80.4984	0.370961
$218$	0	0
$219$	0	0
$220$	−55.0000	−0.250000
$221$	−700.000	−3.16742
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	70.0000	0.312500
$225$	225.000	1.00000
$226$	0	0
$227$	−192.302	−0.847145	−0.423572	−	0.905862i	$-0.639224\pi$
$227$	−192.302	−0.847145	−0.423572	+	0.905862i	$0.639224\pi$
$228$	0	0
$229$	−422.000	−1.84279	−0.921397	−	0.388622i	$-0.872951\pi$
$229$	−422.000	−1.84279	−0.921397	+	0.388622i	$0.872951\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	219.135	0.940492	0.470246	−	0.882535i	$-0.344165\pi$
$233$	219.135	0.940492	0.470246	+	0.882535i	$0.344165\pi$
$234$	−450.000	−1.92308
$235$	0	0
$236$	−102.000	−0.432203
$237$	0	0
$238$	313.050	1.31533
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−270.564	−1.11803
$243$	0	0
$244$	0	0
$245$	−145.000	−0.591837
$246$	0	0
$247$	0	0
$248$	120.748	0.486886
$249$	0	0
$250$	−279.508	−1.11803
$251$	282.000	1.12351	0.561753	−	0.827305i	$-0.310127\pi$
$251$	282.000	1.12351	0.561753	+	0.827305i	$0.310127\pi$
$252$	40.2492	0.159719
$253$	0	0
$254$	70.0000	0.275591
$255$	0	0
$256$	181.000	0.707031
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	111.803	0.430013
$261$	0	0
$262$	0	0
$263$	505.351	1.92149	0.960744	−	0.277436i	$-0.0894847\pi$
$263$	505.351	1.92149	0.960744	+	0.277436i	$0.0894847\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−342.000	−1.27138	−0.635688	−	0.771946i	$-0.719283\pi$
$269$	−342.000	−1.27138	−0.635688	+	0.771946i	$0.719283\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	594.794	2.18674
$273$	0	0
$274$	0	0
$275$	−275.000	−1.00000
$276$	0	0
$277$	−371.187	−1.34003	−0.670013	−	0.742349i	$-0.733712\pi$
$277$	−371.187	−1.34003	−0.670013	+	0.742349i	$0.733712\pi$
$278$	0	0
$279$	162.000	0.580645
$280$	150.000	0.535714
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	−156.525	−0.553091	−0.276546	−	0.961001i	$-0.589190\pi$
$283$	−156.525	−0.553091	−0.276546	+	0.961001i	$0.589190\pi$
$284$	−78.0000	−0.274648
$285$	0	0
$286$	550.000	1.92308
$287$	0	0
$288$	140.872	0.489140
$289$	691.000	2.39100
$290$	0	0
$291$	0	0
$292$	4.47214	0.0153155
$293$	−585.850	−1.99949	−0.999744	−	0.0226391i	$-0.992793\pi$
$293$	−585.850	−1.99949	−0.999744	+	0.0226391i	$0.992793\pi$
$294$	0	0
$295$	−510.000	−1.72881
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−380.000	−1.26246
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	630.000	2.05882
$307$	−478.519	−1.55869	−0.779346	−	0.626594i	$-0.784449\pi$
$307$	−478.519	−1.55869	−0.779346	+	0.626594i	$0.784449\pi$
$308$	−49.1935	−0.159719
$309$	0	0
$310$	−201.246	−0.649181
$311$	402.000	1.29260	0.646302	−	0.763082i	$-0.276315\pi$
$311$	402.000	1.29260	0.646302	+	0.763082i	$0.276315\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	201.246	0.638877
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	205.000	0.640625
$321$	0	0
$322$	0	0
$323$	0	0
$324$	81.0000	0.250000
$325$	559.017	1.72005
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	618.000	1.86707	0.933535	−	0.358487i	$-0.116707\pi$
$331$	618.000	1.86707	0.933535	+	0.358487i	$0.116707\pi$
$332$	165.469	0.498401
$333$	0	0
$334$	−490.000	−1.46707
$335$	0	0
$336$	0	0
$337$	398.020	1.18107	0.590534	−	0.807013i	$-0.298917\pi$
$337$	398.020	1.18107	0.590534	+	0.807013i	$0.298917\pi$
$338$	−740.139	−2.18976
$339$	0	0
$340$	−156.525	−0.460367
$341$	−198.000	−0.580645
$342$	0	0
$343$	−348.827	−1.01699
$344$	−570.000	−1.65698
$345$	0	0
$346$	−530.000	−1.53179
$347$	559.017	1.61100	0.805500	−	0.592596i	$-0.201897\pi$
$347$	559.017	1.61100	0.805500	+	0.592596i	$0.201897\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	−250.000	−0.714286
$351$	0	0
$352$	−172.177	−0.489140
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	−390.000	−1.09859
$356$	2.00000	0.00561798
$357$	0	0
$358$	84.9706	0.237348
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	301.869	0.838525
$361$	361.000	1.00000
$362$	764.735	2.11253
$363$	0	0
$364$	100.000	0.274725
$365$	22.3607	0.0612621
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	702.125	1.88237	0.941187	−	0.337887i	$-0.109712\pi$
$373$	702.125	1.88237	0.941187	+	0.337887i	$0.109712\pi$
$374$	−770.000	−2.05882
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	538.000	1.41953	0.709763	−	0.704441i	$-0.248802\pi$
$379$	538.000	1.41953	0.709763	+	0.704441i	$0.248802\pi$
$380$	0	0
$381$	0	0
$382$	−755.791	−1.97851
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	−245.967	−0.638877
$386$	70.0000	0.181347
$387$	−764.735	−1.97606
$388$	0	0
$389$	−102.000	−0.262211	−0.131105	−	0.991368i	$-0.541853\pi$
$389$	−102.000	−0.262211	−0.131105	+	0.991368i	$0.541853\pi$
$390$	0	0
$391$	0	0
$392$	−194.538	−0.496270
$393$	0	0
$394$	190.000	0.482234
$395$	0	0
$396$	−99.0000	−0.250000
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	−398.020	−1.00005
$399$	0	0
$400$	−475.000	−1.18750
$401$	−782.000	−1.95012	−0.975062	−	0.221931i	$-0.928764\pi$
$401$	−782.000	−1.95012	−0.975062	+	0.221931i	$0.928764\pi$
$402$	0	0
$403$	402.492	0.998740
$404$	0	0
$405$	405.000	1.00000
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−456.158	−1.10450
$414$	0	0
$415$	827.345	1.99360
$416$	350.000	0.841346
$417$	0	0
$418$	0	0
$419$	442.000	1.05489	0.527446	−	0.849588i	$-0.323150\pi$
$419$	442.000	1.05489	0.527446	+	0.849588i	$0.323150\pi$
$420$	0	0
$421$	138.000	0.327791	0.163895	−	0.986478i	$-0.447594\pi$
$421$	138.000	0.327791	0.163895	+	0.986478i	$0.447594\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−782.624	−1.84147
$426$	0	0
$427$	0	0
$428$	−156.525	−0.365712
$429$	0	0
$430$	950.000	2.20930
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	−180.000	−0.414747
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	−368.951	−0.838525
$441$	−261.000	−0.591837
$442$	1565.25	3.54128
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	10.0000	0.0224719
$446$	0	0
$447$	0	0
$448$	183.358	0.409280
$449$	722.000	1.60802	0.804009	−	0.594617i	$-0.202696\pi$
$449$	722.000	1.60802	0.804009	+	0.594617i	$0.202696\pi$
$450$	−503.115	−1.11803
$451$	0	0
$452$	0	0
$453$	0	0
$454$	430.000	0.947137
$455$	500.000	1.09890
$456$	0	0
$457$	−424.853	−0.929656	−0.464828	−	0.885401i	$-0.653884\pi$
$457$	−424.853	−0.929656	−0.464828	+	0.885401i	$0.653884\pi$
$458$	943.621	2.06031
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	−490.000	−1.05150
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	201.246	0.430013
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−684.237	−1.44965
$473$	934.676	1.97606
$474$	0	0
$475$	0	0
$476$	−140.000	−0.294118
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	121.000	0.250000
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	324.230	0.661694
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−495.000	−1.00000
$496$	−342.000	−0.689516
$497$	−348.827	−0.701864
$498$	0	0
$499$	−982.000	−1.96794	−0.983968	−	0.178345i	$-0.942926\pi$
$499$	−982.000	−1.96794	−0.983968	+	0.178345i	$0.942926\pi$
$500$	125.000	0.250000
$501$	0	0
$502$	−630.571	−1.25612
$503$	−353.299	−0.702383	−0.351192	−	0.936304i	$-0.614223\pi$
$503$	−353.299	−0.702383	−0.351192	+	0.936304i	$0.614223\pi$
$504$	270.000	0.535714
$505$	0	0
$506$	0	0
$507$	0	0
$508$	−31.3050	−0.0616239
$509$	138.000	0.271120	0.135560	−	0.990769i	$-0.456717\pi$
$509$	138.000	0.271120	0.135560	+	0.990769i	$0.456717\pi$
$510$	0	0
$511$	20.0000	0.0391389
$512$	212.426	0.414895
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	750.000	1.44231
$521$	−542.000	−1.04031	−0.520154	−	0.854073i	$-0.674125\pi$
$521$	−542.000	−1.04031	−0.520154	+	0.854073i	$0.674125\pi$
$522$	0	0
$523$	−1015.17	−1.94106	−0.970530	−	0.240978i	$-0.922532\pi$
$523$	−1015.17	−1.94106	−0.970530	+	0.240978i	$0.922532\pi$
$524$	0	0
$525$	0	0
$526$	−1130.00	−2.14829
$527$	−563.489	−1.06924
$528$	0	0
$529$	529.000	1.00000
$530$	0	0
$531$	−918.000	−1.72881
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−782.624	−1.46285
$536$	0	0
$537$	0	0
$538$	764.735	1.42144
$539$	319.000	0.591837
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	−490.000	−0.900735
$545$	0	0
$546$	0	0
$547$	1024.12	1.87225	0.936124	−	0.351671i	$-0.114387\pi$
$547$	1024.12	1.87225	0.936124	+	0.351671i	$0.114387\pi$
$548$	0	0
$549$	0	0
$550$	614.919	1.11803
$551$	0	0
$552$	0	0
$553$	0	0
$554$	830.000	1.49819
$555$	0	0
$556$	0	0
$557$	594.794	1.06785	0.533926	−	0.845531i	$-0.320716\pi$
$557$	594.794	1.06785	0.533926	+	0.845531i	$0.320716\pi$
$558$	−362.243	−0.649181
$559$	−1900.00	−3.39893
$560$	−424.853	−0.758666
$561$	0	0
$562$	0	0
$563$	809.457	1.43776	0.718878	−	0.695136i	$-0.244656\pi$
$563$	809.457	1.43776	0.718878	+	0.695136i	$0.244656\pi$
$564$	0	0
$565$	0	0
$566$	350.000	0.618375
$567$	362.243	0.638877
$568$	−523.240	−0.921197
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	−245.967	−0.430013
$573$	0	0
$574$	0	0
$575$	0	0
$576$	369.000	0.640625
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	−1545.12	−2.67322
$579$	0	0
$580$	0	0
$581$	740.000	1.27367
$582$	0	0
$583$	0	0
$584$	30.0000	0.0513699
$585$	1006.23	1.72005
$586$	1310.00	2.23549
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	1140.39	1.93287
$591$	0	0
$592$	0	0
$593$	111.803	0.188539	0.0942693	−	0.995547i	$-0.469949\pi$
$593$	111.803	0.188539	0.0942693	+	0.995547i	$0.469949\pi$
$594$	0	0
$595$	−700.000	−1.17647
$596$	0	0
$597$	0	0
$598$	0	0
$599$	802.000	1.33890	0.669449	−	0.742858i	$-0.266530\pi$
$599$	802.000	1.33890	0.669449	+	0.742858i	$0.266530\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	849.706	1.41147
$603$	0	0
$604$	0	0
$605$	605.000	1.00000
$606$	0	0
$607$	612.683	1.00936	0.504681	−	0.863306i	$-0.331610\pi$
$607$	612.683	1.00936	0.504681	+	0.863306i	$0.331610\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	−281.745	−0.460367
$613$	−943.621	−1.53935	−0.769674	−	0.638437i	$-0.779581\pi$
$613$	−943.621	−1.53935	−0.769674	+	0.638437i	$0.779581\pi$
$614$	1070.00	1.74267
$615$	0	0
$616$	−330.000	−0.535714
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	−918.000	−1.48304	−0.741519	−	0.670932i	$-0.765894\pi$
$619$	−918.000	−1.48304	−0.741519	+	0.670932i	$0.765894\pi$
$620$	90.0000	0.145161
$621$	0	0
$622$	−898.899	−1.44518
$623$	8.94427	0.0143568
$624$	0	0
$625$	625.000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	−450.000	−0.714286
$631$	1042.00	1.65135	0.825674	−	0.564148i	$-0.190795\pi$
$631$	1042.00	1.65135	0.825674	+	0.564148i	$0.190795\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−156.525	−0.246496
$636$	0	0
$637$	−648.460	−1.01799
$638$	0	0
$639$	−702.000	−1.09859
$640$	−771.443	−1.20538
$641$	−302.000	−0.471139	−0.235569	−	0.971858i	$-0.575696\pi$
$641$	−302.000	−0.471139	−0.235569	+	0.971858i	$0.575696\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	543.365	0.838525
$649$	1122.00	1.72881
$650$	−1250.00	−1.92308
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	40.2492	0.0612621
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	442.000	0.668684	0.334342	−	0.942452i	$-0.391486\pi$
$661$	442.000	0.668684	0.334342	+	0.942452i	$0.391486\pi$
$662$	−1381.89	−2.08745
$663$	0	0
$664$	1110.00	1.67169
$665$	0	0
$666$	0	0
$667$	0	0
$668$	219.135	0.328046
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−961.509	−1.42869	−0.714346	−	0.699793i	$-0.753276\pi$
$673$	−961.509	−1.42869	−0.714346	+	0.699793i	$0.753276\pi$
$674$	−890.000	−1.32047
$675$	0	0
$676$	331.000	0.489645
$677$	1346.11	1.98835	0.994175	−	0.107778i	$-0.0343736\pi$
$677$	1346.11	1.98835	0.994175	+	0.107778i	$0.0343736\pi$
$678$	0	0
$679$	0	0
$680$	−1050.00	−1.54412
$681$	0	0
$682$	442.741	0.649181
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	780.000	1.13703
$687$	0	0
$688$	1614.44	2.34657
$689$	0	0
$690$	0	0
$691$	−598.000	−0.865412	−0.432706	−	0.901535i	$-0.642441\pi$
$691$	−598.000	−0.865412	−0.432706	+	0.901535i	$0.642441\pi$
$692$	237.023	0.342519
$693$	−442.741	−0.638877
$694$	−1250.00	−1.80115
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	111.803	0.159719
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	−451.000	−0.640625
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1398.00	−1.97179	−0.985896	−	0.167361i	$-0.946475\pi$
$709$	−1398.00	−1.97179	−0.985896	+	0.167361i	$0.946475\pi$
$710$	872.067	1.22826
$711$	0	0
$712$	13.4164	0.0188433
$713$	0	0
$714$	0	0
$715$	−1229.84	−1.72005
$716$	−38.0000	−0.0530726
$717$	0	0
$718$	0	0
$719$	−718.000	−0.998609	−0.499305	−	0.866427i	$-0.666411\pi$
$719$	−718.000	−0.998609	−0.499305	+	0.866427i	$0.666411\pi$
$720$	−855.000	−1.18750
$721$	0	0
$722$	−807.221	−1.11803
$723$	0	0
$724$	−342.000	−0.472376
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	670.820	0.921457
$729$	729.000	1.00000
$730$	−50.0000	−0.0684932
$731$	2660.00	3.63885
$732$	0	0
$733$	594.794	0.811452	0.405726	−	0.913995i	$-0.367019\pi$
$733$	594.794	0.811452	0.405726	+	0.913995i	$0.367019\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1149.34	1.54689	0.773445	−	0.633864i	$-0.218532\pi$
$743$	1149.34	1.54689	0.773445	+	0.633864i	$0.218532\pi$
$744$	0	0
$745$	0	0
$746$	−1570.00	−2.10456
$747$	1489.22	1.99360
$748$	344.354	0.460367
$749$	−700.000	−0.934579
$750$	0	0
$751$	−478.000	−0.636485	−0.318242	−	0.948009i	$-0.603093\pi$
$751$	−478.000	−0.636485	−0.318242	+	0.948009i	$0.603093\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−1203.00	−1.58708
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	338.000	0.442408
$765$	−1408.72	−1.84147
$766$	0	0
$767$	−2280.79	−2.97365
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	550.000	0.714286
$771$	0	0
$772$	−31.3050	−0.0405505
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	1710.00	2.20930
$775$	450.000	0.580645
$776$	0	0
$777$	0	0
$778$	228.079	0.293161
$779$	0	0
$780$	0	0
$781$	858.000	1.09859
$782$	0	0
$783$	0	0
$784$	551.000	0.702806
$785$	0	0
$786$	0	0
$787$	−621.627	−0.789869	−0.394934	−	0.918709i	$-0.629233\pi$
$787$	−621.627	−0.789869	−0.394934	+	0.918709i	$0.629233\pi$
$788$	−84.9706	−0.107831
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−664.112	−0.838525
$793$	0	0
$794$	0	0
$795$	0	0
$796$	178.000	0.223618
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	391.312	0.489140
$801$	18.0000	0.0224719
$802$	1748.61	2.18031
$803$	−49.1935	−0.0612621
$804$	0	0
$805$	0	0
$806$	−900.000	−1.11663
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	−905.608	−1.11803
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	900.000	1.09890
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	1020.00	1.23487
$827$	−1158.28	−1.40058	−0.700292	−	0.713856i	$-0.746947\pi$
$827$	−1158.28	−1.40058	−0.700292	+	0.713856i	$0.746947\pi$
$828$	0	0
$829$	−1158.00	−1.39686	−0.698432	−	0.715677i	$-0.746118\pi$
$829$	−1158.00	−1.39686	−0.698432	+	0.715677i	$0.746118\pi$
$830$	−1850.00	−2.22892
$831$	0	0
$832$	916.788	1.10191
$833$	907.844	1.08985
$834$	0	0
$835$	1095.67	1.31218
$836$	0	0
$837$	0	0
$838$	−988.342	−1.17941
$839$	1458.00	1.73778	0.868892	−	0.495003i	$-0.164833\pi$
$839$	1458.00	1.73778	0.868892	+	0.495003i	$0.164833\pi$
$840$	0	0
$841$	841.000	1.00000
$842$	−308.577	−0.366481
$843$	0	0
$844$	0	0
$845$	1655.00	1.95858
$846$	0	0
$847$	541.128	0.638877
$848$	0	0
$849$	0	0
$850$	1750.00	2.05882
$851$	0	0
$852$	0	0
$853$	1346.11	1.57809	0.789046	−	0.614334i	$-0.210575\pi$
$853$	1346.11	1.57809	0.789046	+	0.614334i	$0.210575\pi$
$854$	0	0
$855$	0	0
$856$	−1050.00	−1.22664
$857$	−1068.84	−1.24719	−0.623594	−	0.781748i	$-0.714328\pi$
$857$	−1068.84	−1.24719	−0.623594	+	0.781748i	$0.714328\pi$
$858$	0	0
$859$	−438.000	−0.509895	−0.254948	−	0.966955i	$-0.582058\pi$
$859$	−438.000	−0.509895	−0.254948	+	0.966955i	$0.582058\pi$
$860$	−424.853	−0.494015
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	1185.12	1.37008
$866$	0	0
$867$	0	0
$868$	80.4984	0.0927401
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	559.017	0.638877
$876$	0	0
$877$	237.023	0.270266	0.135133	−	0.990827i	$-0.456854\pi$
$877$	237.023	0.270266	0.135133	+	0.990827i	$0.456854\pi$
$878$	0	0
$879$	0	0
$880$	1045.00	1.18750
$881$	−1758.00	−1.99546	−0.997730	−	0.0673435i	$-0.978548\pi$
$881$	−1758.00	−1.99546	−0.997730	+	0.0673435i	$0.978548\pi$
$882$	583.614	0.661694
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	−700.000	−0.791855
$885$	0	0
$886$	0	0
$887$	1578.66	1.77978	0.889890	−	0.456176i	$-0.150781\pi$
$887$	1578.66	1.77978	0.889890	+	0.456176i	$0.150781\pi$
$888$	0	0
$889$	−140.000	−0.157480
$890$	−22.3607	−0.0251244
$891$	−891.000	−1.00000
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−190.000	−0.212291
$896$	−690.000	−0.770089
$897$	0	0
$898$	−1614.44	−1.79782
$899$	0	0
$900$	225.000	0.250000
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1710.00	−1.88950
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	−192.302	−0.211786
$909$	0	0
$910$	−1118.03	−1.22861
$911$	−1742.00	−1.91218	−0.956092	−	0.293066i	$-0.905324\pi$
$911$	−1742.00	−1.91218	−0.956092	+	0.293066i	$0.905324\pi$
$912$	0	0
$913$	−1820.16	−1.99360
$914$	950.000	1.03939
$915$	0	0
$916$	−422.000	−0.460699
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1744.13	−1.88963
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1662.00	−1.78902	−0.894510	−	0.447047i	$-0.852475\pi$
$929$	−1662.00	−1.78902	−0.894510	+	0.447047i	$0.852475\pi$
$930$	0	0
$931$	0	0
$932$	219.135	0.235123
$933$	0	0
$934$	0	0
$935$	1721.77	1.84147
$936$	1350.00	1.44231
$937$	1793.33	1.91390	0.956951	−	0.290249i	$-0.0937381\pi$
$937$	1793.33	1.91390	0.956951	+	0.290249i	$0.0937381\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	1938.00	2.05297
$945$	0	0
$946$	−2090.00	−2.20930
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	100.000	0.105374
$950$	0	0
$951$	0	0
$952$	−939.149	−0.986501
$953$	−782.624	−0.821221	−0.410611	−	0.911811i	$-0.634684\pi$
$953$	−782.624	−0.821221	−0.410611	+	0.911811i	$0.634684\pi$
$954$	0	0
$955$	1690.00	1.76963
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−637.000	−0.662851
$962$	0	0
$963$	−1408.72	−1.46285
$964$	0	0
$965$	−156.525	−0.162202
$966$	0	0
$967$	−1855.94	−1.91927	−0.959636	−	0.281244i	$-0.909253\pi$
$967$	−1855.94	−1.91927	−0.959636	+	0.281244i	$0.909253\pi$
$968$	811.693	0.838525
$969$	0	0
$970$	0	0
$971$	−1622.00	−1.67044	−0.835221	−	0.549914i	$-0.814661\pi$
$971$	−1622.00	−1.67044	−0.835221	+	0.549914i	$0.814661\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	−22.0000	−0.0224719
$980$	−145.000	−0.147959
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	−424.853	−0.431323
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	1106.85	1.11803
$991$	1938.00	1.95560	0.977800	−	0.209539i	$-0.0671965\pi$
$991$	1938.00	1.95560	0.977800	+	0.209539i	$0.0671965\pi$
$992$	281.745	0.284017
$993$	0	0
$994$	780.000	0.784708
$995$	890.000	0.894472
$996$	0	0
$997$	−1372.95	−1.37708	−0.688538	−	0.725200i	$-0.741747\pi$
$997$	−1372.95	−1.37708	−0.688538	+	0.725200i	$0.741747\pi$
$998$	2195.82	2.20022
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	55.3.d.b.54.1	✓	2
3.2	odd	2		495.3.h.b.109.2		2
4.3	odd	2		880.3.i.c.769.1		2
5.2	odd	4		275.3.c.c.76.1		2
5.3	odd	4		275.3.c.c.76.2		2
5.4	even	2	inner	55.3.d.b.54.2	yes	2
11.10	odd	2	inner	55.3.d.b.54.2	yes	2
15.14	odd	2		495.3.h.b.109.1		2
20.19	odd	2		880.3.i.c.769.2		2
33.32	even	2		495.3.h.b.109.1		2
44.43	even	2		880.3.i.c.769.2		2
55.32	even	4		275.3.c.c.76.2		2
55.43	even	4		275.3.c.c.76.1		2
55.54	odd	2	CM	55.3.d.b.54.1	✓	2
165.164	even	2		495.3.h.b.109.2		2
220.219	even	2		880.3.i.c.769.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.3.d.b.54.1	✓	2	1.1	even	1	trivial
55.3.d.b.54.1	✓	2	55.54	odd	2	CM
55.3.d.b.54.2	yes	2	5.4	even	2	inner
55.3.d.b.54.2	yes	2	11.10	odd	2	inner
275.3.c.c.76.1		2	5.2	odd	4
275.3.c.c.76.1		2	55.43	even	4
275.3.c.c.76.2		2	5.3	odd	4
275.3.c.c.76.2		2	55.32	even	4
495.3.h.b.109.1		2	15.14	odd	2
495.3.h.b.109.1		2	33.32	even	2
495.3.h.b.109.2		2	3.2	odd	2
495.3.h.b.109.2		2	165.164	even	2
880.3.i.c.769.1		2	4.3	odd	2
880.3.i.c.769.1		2	220.219	even	2
880.3.i.c.769.2		2	20.19	odd	2
880.3.i.c.769.2		2	44.43	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 55 = 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	55.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.23607 q^{2} +1.00000 q^{4} +5.00000 q^{5} +4.47214 q^{7} +6.70820 q^{8} +9.00000 q^{9} -11.1803 q^{10} -11.0000 q^{11} +22.3607 q^{13} -10.0000 q^{14} -19.0000 q^{16} -31.3050 q^{17} -20.1246 q^{18} +5.00000 q^{20} +24.5967 q^{22} +25.0000 q^{25} -50.0000 q^{26} +4.47214 q^{28} +18.0000 q^{31} +15.6525 q^{32} +70.0000 q^{34} +22.3607 q^{35} +9.00000 q^{36} +33.5410 q^{40} -84.9706 q^{43} -11.0000 q^{44} +45.0000 q^{45} -29.0000 q^{49} -55.9017 q^{50} +22.3607 q^{52} -55.0000 q^{55} +30.0000 q^{56} -102.000 q^{59} -40.2492 q^{62} +40.2492 q^{63} +41.0000 q^{64} +111.803 q^{65} -31.3050 q^{68} -50.0000 q^{70} -78.0000 q^{71} +60.3738 q^{72} +4.47214 q^{73} -49.1935 q^{77} -95.0000 q^{80} +81.0000 q^{81} +165.469 q^{83} -156.525 q^{85} +190.000 q^{86} -73.7902 q^{88} +2.00000 q^{89} -100.623 q^{90} +100.000 q^{91} +64.8460 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 10 q^{5} + 18 q^{9} - 22 q^{11} - 20 q^{14} - 38 q^{16} + 10 q^{20} + 50 q^{25} - 100 q^{26} + 36 q^{31} + 140 q^{34} + 18 q^{36} - 22 q^{44} + 90 q^{45} - 58 q^{49} - 110 q^{55} + 60 q^{56}+ \cdots - 198 q^{99}+O(q^{100}) \) 2 * q + 2 * q^4 + 10 * q^5 + 18 * q^9 - 22 * q^11 - 20 * q^14 - 38 * q^16 + 10 * q^20 + 50 * q^25 - 100 * q^26 + 36 * q^31 + 140 * q^34 + 18 * q^36 - 22 * q^44 + 90 * q^45 - 58 * q^49 - 110 * q^55 + 60 * q^56 - 204 * q^59 + 82 * q^64 - 100 * q^70 - 156 * q^71 - 190 * q^80 + 162 * q^81 + 380 * q^86 + 4 * q^89 + 200 * q^91 - 198 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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